Editing Cube (section)

== Related figures ==
=== Construction of polyhedra ===
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The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following:
* When [[faceting]] a cube, meaning removing part of the polygonal faces without creating new vertices of a cube, the resulting polyhedron is the [[stellated octahedron]].{{r|inchbald}}
* The cube is [[non-composite polyhedron]], meaning it is a convex polyhedron that cannot be separated into two or more regular polyhedra. The cube can be applied to construct a new convex polyhedron by attaching another.{{r|timofeenko-2010}} Attaching a [[square pyramid]] to each square face of a cube produces its [[Kleetope]], a polyhedron known as the [[tetrakis hexahedron]].{{r|sod}} Suppose one and two equilateral square pyramids are attached to their square faces. In that case, they are the construction of an [[elongated square pyramid]] and [[elongated square bipyramid]] respectively, the [[Johnson solid]]'s examples.{{r|rajwade}}
* Each of the cube's vertices can be [[Truncation (geometry)|truncated]], and the resulting polyhedron is the [[Archimedean solid]], the [[truncated cube]].{{sfnp|Cromwell|1997|pp=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA81 81–82]}} When its edges are truncated, it is a [[rhombicuboctahedron]].{{r|linti}} Relatedly, the rhombicuboctahedron can also be constructed by separating the cube's faces and then spreading away, after which adding other triangular and square faces between them; this is known as the "expanded cube". Similarly, it is constructed by the cube's dual, the regular octahedron.{{r|vxac}}
* The [[barycentric subdivision]] of a cube (or its dual, the regular octahedron) is the [[disdyakis dodecahedron]], a [[Catalan solid]].{{r|ls}}
* The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in a [[trirectangular tetrahedron]].{{sfnp|Coxeter|1973|p=[http://books.google.com/books?id=2ee7AQAAQBAJ&pg=PA71 71]}}
* The [[snub cube]] is an Archimedean solid that can be constructed by separating away the cube square's face, and filling their gaps with twisted angle equilateral triangles, a process known as [[Snub (geometry)|snub]].{{r|holme}}
The cube can be constructed with six [[square pyramid]]s, tiling space by attaching their apices. In some cases, this produces the [[rhombic dodecahedron]] circumscribing a cube.{{r|barnes|cundy}}

=== Polycubes ===
{{main|Polycubes}}
[[File:Net of tesseract.gif|thumb|upright=0.6|[[Dali cross]], the net of a [[tesseract]]]]
[[Polycube]] is a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as the [[polyominoes]] in three-dimensional space.{{r|lunnon}} When four cubes are stacked vertically, and the other four are attached to the second-from-top cube of the stack, the resulting polycube is [[Dali cross]], after [[Salvador Dali]]. In addition to popular cultures, the Dali cross is a tile space polyhedron,{{r|hut|pucc}} which can be represented as the net of a [[tesseract]]. A tesseract is a cube analogous' [[four-dimensional space]] bounded by twenty-four squares and eight cubes.{{r|hall}}

=== Space-filling and honeycombs ===
[[Hilbert's third problem]] asked whether every two equal-volume polyhedra could always be dissected into polyhedral pieces and reassembled into each other. If it were, then the volume of any polyhedron could be defined axiomatically as the volume of an equivalent cube into which it could be reassembled. [[Max Dehn]] solved this problem in an invention [[Dehn invariant]], answering that not all polyhedra can be reassembled into a cube.{{r|gruber}} It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different.{{r|zeeman}}

[[File:Partial cubic honeycomb.png|thumb|upright=0.6|[[Cubic honeycomb]]]]
The cube has a Dehn invariant of zero. This indicates the cube is applied for [[Honeycomb (geometry)|honeycomb]]. More strongly, the cube is a [[Space-filling polyhedron|space-filling tile]] in three-dimensional space in which the construction begins by attaching a polyhedron onto its faces without leaving a gap.{{r|lm}} The cube is a [[plesiohedron]], a special kind of space-filling polyhedron that can be defined as the [[Voronoi cell]] of a symmetric [[Delone set]].{{r|erdahl}} The plesiohedra include the [[parallelohedra]], which can be [[Translation (geometry)|translated]] without rotating to fill a space in which each face of any of its copies is attached to a like face of another copy. There are five kinds of parallelohedra, one of which is the cuboid.{{r|alexandrov}} Every three-dimensional parallelohedron is [[zonohedron]], a [[centrally symmetric]] polyhedron whose faces are [[Zonogon|centrally symmetric polygons]].{{r|shephard}} In the case of cube, it can be represented as the [[Cell (geometry)|cell]]. Some honeycombs have cubes as the only cells; one example is [[cubic honeycomb]], the only regular honeycomb in Euclidean three-dimensional space, having four cubes around each edge.{{r|twelveessay|ns}}

=== Miscellaneous ===
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 | footer = Enumeration according to {{harvtxt|Skilling|1976}}: compound of six cubes with rotational freedom <math> \mathrm{UC}_7 </math>, [[Compound of three cubes|three cubes]] <math> \mathrm{UC}_8 </math>, and [[Compound of five cubes|five cubes]] <math> \mathrm{UC}_9 </math>
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{{anchor|Compound of cubes}}Compound of cubes is the [[polyhedral compound]]s in which the cubes share the same centre. They belong to the [[uniform polyhedron compound]], meaning they are polyhedral compounds whose constituents are identical (although possibly [[enantiomorphous]]) [[uniform polyhedra]], in an arrangement that is also uniform. Respectively, the list of compounds enumerated by {{harvtxt|Skilling|1976}} in the seventh to ninth uniform compounds for the compound of six cubes with rotational freedom, [[Compound of three cubes|three cubes]], and five cubes.{{r|skilling}} Two compounds, consisting of [[compound of two cubes|two]] and three cubes were found in [[M. C. Escher|Escher]]'s [[wood engraving]] print [[Stars (M. C. Escher)|''Stars'']] and [[Max Brückner]]'s book ''Vielecke und Vielflache''.{{r|hart}}

{{multiple image
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 | caption1 = Spherical cube
 | image2 = 3-Manifold 3-Torus.png
 | caption2 = A view in [[3-torus|three-dimensional torus]]
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{{anchor|Spherical cube}}The spherical cube represents the [[spherical polyhedron]], which can be modeled by the [[Arc (geometry)|arc]] of [[great circle]]s, creating bounds as the edges of a [[spherical polygon|spherical square]].{{r|yackel}}
Hence, the spherical cube consists of six spherical squares with 120° interior angles on each vertex. It has [[vector equilibrium]], meaning that the distance from the centroid and each vertex is the same as the distance from that and each edge.{{r|popko|fuller}} Its dual is the [[spherical octahedron]].{{r|yackel}}

The topological object [[3-torus|three-dimensional torus]] is a topological space defined to be [[homeomorphic]] to the Cartesian product of three circles. It can be represented as a three-dimensional model of the cube shape.{{r|marar}}